Equivariant Embeddings of Algebraic Groups

with the unique extension of the function f(g · x 0 ) := f(g) to X.
Corollary 3. Let X be an affine G-embedding with base point x 0 . If x ′ 0 = hx 0 is another base
point, then
A Γ(X,h·x0 ) = r h (A Γ(X,x0 )), (3.10)
where r h denotes right translation by h in k[G], r h (f)(x) = f(xh).
Proof. Suppose X is an affine G-embedding.
We have shown in (3.7) that the set Γ(X, x 0 ) is
determined by X only up to conjugation, as any other base point is of the form h · x 0 for a unique
element h ∈ G and Γ(X, h · x 0 ) = hΓ(X, x 0 )h −1 . We now show that A Γ(X,h·x0 ) = r h (A Γ(X,x0 )).
Suppose f ∈ r h (A Γ(X,x0 )). Then f = r h (f ′ ) for some f ′ ∈ A Γ(X,x0 ), which means that v γ (f ′ ) ≥ 0
for all γ ∈ Γ(X, x 0 ). If γ ′ ∈ Γ(X, h · x 0 ) = hΓ(X, x 0 )h −1 , write γ ′ = h • γ for γ ∈ Γ(X, x 0 ) and
consider
v h•γ (f) = inf
s∈U v t(f(s · hγ(t)h −1 )) = inf
s∈U v t((r h f ′ )(s · hγ(t)h −1 )) = inf
s∈U v t(f ′ (s · hγ(t)h −1 · h))
= inf v t(f ′ (sh · γ(t))) = inf v t(f ′ (s ′ · γ(t))) = v γ (f ′ ) ≥ 0.
s∈U s ′ ∈Uh
Therefore, f ∈ A Γ(X,h·x0 ), so r h (A Γ(X,x0 )) ⊂ A Γ(X,h·x0 ). Likewise, r h −1(A Γ(X,h·x0 )) ⊂ A Γ(X,x0 ), so
r h (A Γ(X,x0 )) = A Γ(X,h·x0 ) as claimed.
Consider the following examples.
Example 9 (Recovering G from Γ(G, e)). Recall that in Example 5 we saw Γ(G, e) = {ε} for
any group G. Now A {ε} = {f ∈ k[G] : v ε (f) ≥ 0} = k[G], and G = Spec k[G] = Spec A {ε} . Note
that Γ(G, e) = {ε} is G-stable for conjugation and that A {ε} is right-invariant as well. This will be
true in general. See Section 3.3.2 for more details.
Example 10 (Recovering M n from Γ(M n , I n )). Recall Example 8, the embedding of G =
{ ( t p 1 0
GL n in X = M n . Here Γ(M n , I n ) = A . .. A −1 : A ∈ GL n , p 1 , . . . , p n ∈ N 0
}, whose
0 t pn )
associated algebra is A Γ(Mn,I n) = {f ∈ k[GL n ] : v γ (f)) ≥ 0 for all γ ∈ Γ(M n , I n )}. This is naturally
isomorphic to k[X 11 , . . . , X nn ] ⊂ k[GL n ] = k[X 11 , . . . , X nn , det −1 ], and clearly Spec A Γ(Mn,I n) =
M n . Again we see that Γ is GL n -stable for conjugation, and that the corresponding embedding M n
is biequivariant.
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